The Gravitational Force Between The Sun (mass $=1.99 \times 10^{30} \, \text{kg}$) And Mercury (mass $=3.30 \times 10^{23} \, \text{kg}$) Is $8.99 \times 10^{21} \, \text{N}$. How Far Is Mercury From The Sun?A. $6.98

Introduction

The gravitational force between two objects is a fundamental concept in physics, governed by the law of universal gravitation. This law, formulated by Sir Isaac Newton, describes the attractive force between two masses as being proportional to the product of their masses and inversely proportional to the square of the distance between their centers. In this article, we will apply this law to calculate the distance between the Sun and Mercury, given the gravitational force between them.

The Law of Universal Gravitation

The law of universal gravitation is expressed mathematically as:

$$F = G \frac{m_1 m_2}{r^2}$$

where:

• $F$ is the gravitational force between the two objects
• $G$ is the gravitational constant ($6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2 \cdot \text{kg}^{-2}$)
• $m_1$ and $m_2$ are the masses of the two objects
• $r$ is the distance between the centers of the two objects

Given Values

We are given the following values:

• Mass of the Sun: $m_1 = 1.99 \times 10^{30} \, \text{kg}$
• Mass of Mercury: $m_2 = 3.30 \times 10^{23} \, \text{kg}$
• Gravitational force between the Sun and Mercury: $F = 8.99 \times 10^{21} \, \text{N}$

Calculating the Distance

We can now use the law of universal gravitation to calculate the distance between the Sun and Mercury. Rearranging the equation to solve for $r$, we get:

$$r = \sqrt{\frac{G m_1 m_2}{F}}$$

Plugging in the given values, we get:

$$r = \sqrt{\frac{(6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2 \cdot \text{kg}^{-2})(1.99 \times 10^{30} \, \text{kg})(3.30 \times 10^{23} \, \text{kg})}{8.99 \times 10^{21} \, \text{N}}}$$

Simplifying the expression, we get:

$$r = \sqrt{\frac{6.674 \times 10^{-11} \times 1.99 \times 10^{30} \times 3.30 \times 10^{23}}{8.99 \times 10^{21}}}$$

$$r = \sqrt{\frac{4.38 \times 10^{43}}{8.99 \times 10^{21}}}$$

$$r = \sqrt{4.87 \times 10^{21}}$$

$$r = 6.98 \times 10^{10} \, \text{m}$$

Conclusion

In this article, we applied the law of universal gravitation to calculate the distance between the Sun and Mercury, given the gravitational force between them. We found that the distance between the Sun and Mercury is approximately $6.98 \times 10^{10} \, \text{m}$.

Discussion

The law of universal gravitation is a fundamental concept in physics that describes the attractive force between two masses. This law has been extensively tested and validated through various experiments and observations. The calculation of the distance between the Sun and Mercury using the law of universal gravitation is a classic example of how this law can be applied to real-world problems.

Limitations

One limitation of this calculation is that it assumes a spherical shape for the Sun and Mercury. In reality, the shapes of these objects are more complex and may affect the gravitational force between them. Additionally, this calculation does not take into account the effects of other celestial bodies on the gravitational force between the Sun and Mercury.

Future Work

Future work could involve calculating the gravitational force between the Sun and other planets in our solar system, using the law of universal gravitation. This could provide valuable insights into the dynamics of our solar system and the effects of gravitational forces on the motion of celestial bodies.

References

• Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica.
• Feynman, R. P. (1963). The Feynman Lectures on Physics.
• Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics.
  The Gravitational Force Between the Sun and Mercury: Q&A ===========================================================

Q: What is the law of universal gravitation?

A: The law of universal gravitation is a fundamental concept in physics that describes the attractive force between two masses. It was formulated by Sir Isaac Newton and is expressed mathematically as:

$$F = G \frac{m_1 m_2}{r^2}$$

where:

• $F$ is the gravitational force between the two objects
• $G$ is the gravitational constant ($6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2 \cdot \text{kg}^{-2}$)
• $m_1$ and $m_2$ are the masses of the two objects
• $r$ is the distance between the centers of the two objects

Q: What is the gravitational constant?

A: The gravitational constant, denoted by $G$, is a fundamental constant of nature that describes the strength of the gravitational force between two objects. Its value is approximately $6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2 \cdot \text{kg}^{-2}$.

Q: How does the gravitational force between two objects depend on their masses?

A: The gravitational force between two objects depends on the product of their masses. The more massive the objects, the stronger the gravitational force between them.

Q: How does the gravitational force between two objects depend on the distance between them?

A: The gravitational force between two objects depends on the inverse square of the distance between them. The farther apart the objects, the weaker the gravitational force between them.

Q: What is the distance between the Sun and Mercury?

A: The distance between the Sun and Mercury is approximately $6.98 \times 10^{10} \, \text{m}$.

Q: How was the distance between the Sun and Mercury calculated?

A: The distance between the Sun and Mercury was calculated using the law of universal gravitation, which describes the attractive force between two masses. The law is expressed mathematically as:

$$F = G \frac{m_1 m_2}{r^2}$$

where:

• $F$ is the gravitational force between the two objects
• $G$ is the gravitational constant ($6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2 \cdot \text{kg}^{-2}$)
• $m_1$ and $m_2$ are the masses of the two objects
• $r$ is the distance between the centers of the two objects

Q: What are some limitations of this calculation?

A: One limitation of this calculation is that it assumes a spherical shape for the Sun and Mercury. In reality, the shapes of these objects are more complex and may affect the gravitational force between them. Additionally, this calculation does not take into account the effects of other celestial bodies on the gravitational force between the Sun and Mercury.

Q: What are some potential applications of this calculation?

A: This calculation has potential applications in the fields of astronomy and astrophysics, where understanding the gravitational forces between celestial bodies is crucial for predicting their motion and behavior.

Q: What are some future directions for research in this area?

A: Future research in this area could involve calculating the gravitational force between the Sun and other planets in our solar system, using the law of universal gravitation. This could provide valuable insights into the dynamics of our solar system and the effects of gravitational forces on the motion of celestial bodies.